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### Why is this differential injective? Lee Smooth Manifolds Proposition 5.3

This is true in general: If $f\colon X \to Y$ and $g\colon Y \to X$ are two functions satisfying $$g\circ f = \operatorname{id}_X,$$ then $f$ is injective. If $x,y \in X$ are such that $f(x) = f(y)$,...
• 1,466
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• 5,991
1 vote
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• 12.3k
1 vote
Accepted

### Use of intermediate value theorem to show that $p = (\cos 2\pi x, \sin 2 \pi x)$ is a covering map for $S^1$.

As Munkres writes, the fact that $p$ is a covering map comes from elementary properties of the sine and cosine functions. Which properties do we need? $\sin$ is strictly monotonically increasing on ...
• 75.2k
1 vote
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### "Converse" to Chinese Remainder Theorem

First, this is only true because $q_1$ and $q_2$ are coprime $\iff (q_1,q_2)=1$: Suppose \$\color{red}a \equiv \color{blue}{a_1}q_2+\color{green}{a_2}q_1\equiv \color{blue}{a_3}q_2+\color{green}{a_4}...
• 3,920

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